Fourier analytic techniques in geometry and analysis

A powerful discovery of Joseph Fourier in the early 1800s was that certain functions could be written as an infinite sum of simple 'wave-like functions'. Such a decomposition is now known as a Fourier series, and has had wide-ranging applications across mathematics and wider science, for example in signal processing and in solving complicated differential equations. The Fourier transform describes how quickly the Fourier series converges, i.e. the decay rate of the amplitudes of the waves in the decomposition as frequency increases. One way of viewing this is that the faster the Fourier transform decays, the more wave-like the function was to begin with. Thus, some geometric information about the original object is captured by Fourier decay. This research project considers the Fourier transform of measures (mass distributions), which are analogous to functions. It is well known that the Fourier transform of a measure encodes a lot of information about its geometric structure, for example concerning its dimension, curvature properties, and arithmetic resonances. We investigate the Fourier transform, and the geometric information it encodes, in several challenging contexts. For example, we consider how it is affected when the original measure is distorted under standard geometric operations, such as projecting a measure in 2 dimensional space onto lines. Similar questions about the Hausdorff dimension of sets and measures are at the heart of geometric measure theory and we will establish Fourier analytic analogues of classical results in this direction. We also consider the Fourier transform in probabilistic settings. Brownian motion is a fundamental random process - first observed as the seemingly random path a grain of pollen follows when suspended in water - and is our archetypal example. We will consider the Fourier transform of natural (random) measures associated with Brownian motion and related processes. Finally, we will consider dynamically invariant measures, where we will use transfer operators and other tools from the thermodynamic formalism to analyse the Fourier decay.

The main source of impact from this project will be academic, however, there are also several promising avenues for indirect impact and pathways to impact, for example, through the PI and his passion for public engagement and education. In order to accelerate the impact of the work, the findings of this project will be made available online via the arXiv and links to these articles will be posted on the PI and Co-I's personal homepages at the University of St Andrews as well as the University's online research repository, ensuring open access. The work will also be submitted to peer reviewed journals of the highest standard and the findings will be presented at both national and international conferences and seminars.
The PI will ensure that he is well-placed to act upon any avenues for wider societal impact which arise during the project. This will be achieved through training and interaction. On the training side, in 2017 the PI became the first pure mathematician to be awarded a place on the highly competitive Scottish Crucible, which is an intensive leadership and development programme designed to stimulate collaboration and accelerate impact outside the academic community. On the interaction side, the PI will liaise with the University of St Andrews Knowledge Transfer Centre, which is dedicated to increasing the wider use and positive impact of the University's research activities. The Fourier transform, which is the central object of study in this project, has had an enormous amount of impact on society and the economy over the last 100 years. For example, it allows a complex signal to be broken down into its constituent frequencies (a Fourier series), which has wide-ranging applications in signal processing and time series analysis. This project will further develop the mathematics behind the Fourier transform, which will contribute to the continued exploration into applications and future opportunities for impact.
The PI is well-equipped to disseminate the findings of this research project to both the academic community and wider society over the coming years. He has taken part in numerous outreach and engagement activities and has won several awards for his teaching. The Co-I also has a strong reputation for outreach and engagement. He is active in the public promotion of mathematics and has given many talks to science festivals and school audiences.
Work from this project will contribute to some of the PI and Co-I's future endeavours outside research, for example, in teaching advanced undergraduate courses, running summer schools, public engagement (science festivals etc), and in training future research students and PDRAs. In July 2017 the PI will give an invited course on fractal geometry at the LMS Undergraduate Summer School, which will be hosted by the University of Manchester. The course is aimed at top undergraduates from across the country and will involve aspects of the PI's active research projects. The PI was recently invited to participate in the 2018 British Congress of Mathematics Education and, together with Tom Kempton (Manchester), the PI is planning to submit a proposal to the Edinburgh Science Festival to give a workshop entitled `Seven big ideas in mathematics', which will attempt to communicate some cutting edge ideas from across mathematics to a general audience.
Finally, this research project will provide a platform for stimulating future impact. There are several talented early career researchers currently working on fractal geometry in the UK and abroad and the PI sees this as an excellent opportunity to foster the next generation of researchers in the field. The Co-I has a great deal of experience in supervising PDRAs and PhD students, which will be extremely beneficial to the project. The 2015 EPSRC Strategic Plan highlights the desire to `nurture the next generation of skilled researchers and innovators' and this is a central part of our short and long term vision.